The second stack is the tropical universal Jacobian over \(\operatorname{\bf {LSch}}\), denoted by \(\widetilde{\mathcal {J}}^{trop}_{g,n}\) , whose fiber over \(S\in \operatorname{\bf {LSch}}\)  is the groupoid whose objects consists of a pair \((\Gamma _s/\overline{M}_{S,s},D_s)\)  for each geometric point \(s\)  of \(\underline{S}\) , where \(\Gamma _s/\overline{M}_{S,s}\)  is a quasi-stable tropical curve over the monoid \(\overline{M}_{S,s}\)  of type \((g,n)\)  and \(D_s\)  is an admissible divisor on \((\Gamma _s)\) , subject to a natural compatibility relation with respect to étale specializations of geometric points of \(\underline{S}\) , see Definition REF . In Proposition REF , we prove that \(\widetilde{\mathcal {J}}^{trop}_{g,n}\)  is the Artin fan (hence in particular a logarithmic algebraic stack) associated to the cone stack \(\mathcal {J}^{trop}_{g,n}\)  under the natural equivalence of 2-categories between cone stacks and Artin fans over \(k\)  (see [1]}).
